No, it means they are very distantly related and have shared ancestors many generations ago. First cousins share a set of grandparents. Second cousins share set of great-grandparents. Third cousins share a set of great-great-grandparents, and so on.

To add to Susan612's answer, many U.S. native speakers are vague about the meaning of "once removed" unless they make hobby of genealogy (tracing family ancestors). It refers to a difference in generations. First cousins share grandparents and thus are in the same "generation." Let's say that Lois is my mother's sister's daughter. "Lois and I are first cousins." Lois and I are in the same generation and roughly the same age. Lois has a son named Scott. Scott is not in the same generation as me, he is one generation younger. Scott is my "first cousin once removed." Scott's baby daughter would be my "first cousin twice removed." To say that Jennifer Lawrence is Taylor's 16th cousin twice removed is almost a joke. Very few people in the United States know or care about a relationship that distant.